Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and graph the solution on a real number line. \(-6 \leq 3 x-10<6\)

Step-by-Step Solution

Answer

The solution to the inequality is \( \frac{4}{3} \leq x < \frac{16}{3} \). This solution is represented on a real number line as an interval starting from 4/3 (inclusive) up to 16/3 (exclusive)

Step 1: Add 10 on both sides of inequality

The first step is to get the algebraic part, \(3x - 10\), isolated. This can be achieved by adding 10 to all parts of this inequality, which will help to get rid of -10. So, \( -6 + 10 \leq 3x -10 + 10 < 6 + 10 \) will become \(4 \leq 3x < 16\).

Step 2: Divide by 3 on both sides of inequality

Next, to isolate the variable x, divide all parts of the inequality by 3. This transforms the inequality into \( \frac{4}{3} \leq x < \frac{16}{3}\) . Thus, values of x can be any real number that falls within the closed and open interval \([\frac{4}{3}, \frac{16}{3})\)

Step 3: Graph the solution on a real number line

On a real number line, mark the point \(\frac{4}{3}\) with a closed circle (indicating that this value is included in the solution) and the point \(\frac{16}{3}\) with an open circle (indicating that this value is not part of the solution). Then, draw a line connecting these two circles. Any point on this line represents a possible value of x.

Key Concepts

Real Numbers

Real numbers include all the numbers you can think of, such as whole numbers, fractions, and decimals. They form a continuous line without any gaps, making them an essential part of math. When solving inequalities, we often deal with real numbers, since we want to find a range of possible values for a variable.

These numbers stretch infinitely in both directions on the number line, including everything from negative infinity, through zero, to positive infinity. Understanding real numbers is crucial when working on problems involving inequalities because solutions are often represented as a segment of the real number line.

Algebraic Expressions

Algebraic expressions are a combination of numbers and variables, often joined by operations like addition, subtraction, multiplication, and division. In the inequality \(-6 \leq 3x-10<6\), the expression \((3x - 10)\) is an algebraic expression.

Solving such inequalities involves manipulating these expressions to isolate the variable (usually represented by \(x\)) on one side.

• First, we perform operations to both sides to maintain the balance of the inequality.
• Then, we simplify the expression to find the range of values the variable can take.

Mastering algebraic expressions is key to solving inequalities efficiently.

Solution Set

The solution set of an inequality includes all values that satisfy the given inequality. When we solved \(-6 \leq 3x-10<6\), we found that \(x\) can take any value between \(\frac{4}{3}\) and \(\frac{16}{3}\).

This range is known as the solution set. Solution sets can vary in form, from a single value to a range of numbers. To identify them:

• Solve the inequality step by step to isolate the variable.
• Determine the conditions under which the inequality holds true.

Knowing how to define and interpret solution sets is vital for understanding the scope of any mathematical problem.

Interval Notation

Interval notation is a way of expressing the solution set of an inequality using specific symbols to denote the range of possible values. In our example, the solution \([\frac{4}{3}, \frac{16}{3})\) uses interval notation.

Here's how to read it:

• The square bracket \([\) indicates that \(\frac{4}{3}\) is included in the solution set.
• The parenthesis \(()\) means that \(\frac{16}{3}\) is not included.

Understanding and using interval notation makes it easier to communicate mathematical ideas clearly and concisely.